I hear & read the term "anecdotal evidence" a great deal. But wouldn't it be better to say "anecdotal illustration"? After all, most of the time a few anecdotes do nothing to increase confidence in a hypothesis, rather, they illustrate possible instances assuming that the hypothesis is true.
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Anecdotal evidence is what you hang people on. Heaps of duff temperature measurements and inadequate mathematical models is what you pass off as science.
The term "evidence" - even the term "relevant evidence" the way lawyers use it (scroll to rule 401 here) - is a pretty broad term and can include "weakly supportive evidence." The word "anecdotal" puts the listener on notice that the speaker acknowledges the limitations of this evidence, so I think the term is useful as it stands.
In some areas (like the political polling example you used a little below), something much better is available, and the anecdotes are useless. In others, anecdotal evidence is all there is.
In the extreme case, rejection of anecdotal evidence leads people to ignore research possibilities. For example, when Alexander Fleming saw that the penicillin mold had killed the bacteria on one of his plates, he could have ignored it as inconsequential. There are a lot of one-time observations that, if followed up on, might lead to interesting results.
The criteria for an interesting line of investigation, in other words, are necessaily and by definition more lenient, or at least different, than the criteria for a valid theory. If you're too fussy when you choose lies of research, you're less likely to discover anything big.